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Properties

Label 2-369600-1.1-c1-0-139
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 11-s − 13-s − 3·17-s + 21-s + 3·23-s + 27-s − 5·29-s − 31-s + 33-s − 4·37-s − 39-s − 3·41-s + 5·43-s + 6·47-s + 49-s − 3·51-s + 9·53-s + 59-s − 3·61-s + 63-s − 4·67-s + 3·69-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.727·17-s + 0.218·21-s + 0.625·23-s + 0.192·27-s − 0.928·29-s − 0.179·31-s + 0.174·33-s − 0.657·37-s − 0.160·39-s − 0.468·41-s + 0.762·43-s + 0.875·47-s + 1/7·49-s − 0.420·51-s + 1.23·53-s + 0.130·59-s − 0.384·61-s + 0.125·63-s − 0.488·67-s + 0.361·69-s − 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.263489041\)
\(L(\frac12)\) \(\approx\) \(3.263489041\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.55515828514129, −12.00670727440653, −11.70900599269344, −10.93465057184055, −10.84819860104623, −10.27754231587291, −9.661137025842434, −9.234905341938197, −8.911072233735219, −8.475070431841193, −7.934192368077148, −7.440869801143661, −7.030403685334210, −6.667508585006813, −5.925900745862434, −5.522509852992117, −4.895487492221338, −4.499659984170647, −3.870146122263821, −3.529746992117289, −2.821671411241776, −2.263592888171601, −1.862160966540288, −1.154430811423675, −0.4587535765296710, 0.4587535765296710, 1.154430811423675, 1.862160966540288, 2.263592888171601, 2.821671411241776, 3.529746992117289, 3.870146122263821, 4.499659984170647, 4.895487492221338, 5.522509852992117, 5.925900745862434, 6.667508585006813, 7.030403685334210, 7.440869801143661, 7.934192368077148, 8.475070431841193, 8.911072233735219, 9.234905341938197, 9.661137025842434, 10.27754231587291, 10.84819860104623, 10.93465057184055, 11.70900599269344, 12.00670727440653, 12.55515828514129

Graph of the $Z$-function along the critical line